Yes. A simple argument is that $BO \to BTOP$ is a rational equivalence and an H-space map (in fact even an infinite loop map), so it follows from the Whitney sum formula for vector bundles.
Edit: The argument for this is as follows. Let $\mu : BTOP \times BTOP \to BTOP$ be the map corresponding to Whitney sum of (stable, topological) bundles. The question is whether the identity $$\mu^* p_n = \sum_{ a + b = n} p_a \otimes p_b$$ holds. As the map $BO \to BTOP$ is a rational equivalence and an H-space map, it is equivalent to verify this equation in the cohomology of $BO$ instead, where it indeed holds.